Earthquake Cycle Simulations with Rate-And-State Friction and Power-Law T Viscoelasticity

Tectonophysics 733 (2018) 232–256

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Tectonophysics

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Earthquake cycle simulations with rate-and-state friction and power-law T viscoelasticity

Kali L. Allisona,*, Eric M. Dunhama,b a Department of Geophysics, Stanford University, Stanford, CA, USA b Institute for Computational and Mathematical Engineering, Stanford University, Stanford, CA, USA

ARTICLE INFO ABSTRACT

Keywords: We simulate earthquake cycles with rate-and-state fault friction and off-fault power-law viscoelasticity for the Earthquake cycle classic 2D antiplane shear problem of a vertical, strike-slip plate boundary fault. We investigate the interaction Viscoelastic flow between fault slip and bulk viscous flow with experimentally-based flow laws for quartz-diorite and olivine for Power-law rheology the crust and mantle, respectively. Simulations using three linear geotherms (dT/dz=20, 25, and 30 K/km) Strike-slip produce different deformation styles at depth, ranging from significant interseismic fault creep to purely bulk Brittle-ductile transition viscous flow. However, they have almost identical earthquake recurrence interval, nucleation depth, and down- Rate-and-state friction Summation-by-parts operators dip coseismic slip limit. Despite these similarities, variations in the predicted surface deformation might permit discrimination of the deformation mechanism using geodetic observations. Additionally, in the 25 and 30 K/km simulations, the crust drags the mantle; the 20 K/km simulation also predicts this, except within 10 km of the fault where the reverse occurs. However, basal tractions play a minor role in the overall force balance of the lithosphere, at least for the flow laws used in our study. Therefore, the depth-integrated stress on the fault is balanced primarily by shear stress on vertical, fault-parallel planes. Because strain rates are higher directly below the fault than far from it, stresses are also higher. Thus, the upper crust far from the fault bears a substantial part of the tectonic load, resulting in unrealistically high stresses. In the real Earth, this might lead to distributed plastic deformation or formation of subparallel faults. Alternatively, fault pore pressures in excess of hydrostatic and/or weakening mechanisms such as grain size reduction and thermo-mechanical coupling could lower the strength of the ductile fault root in the lower crust and, concomitantly, off-fault upper crustal stresses.

1. Introduction more nuanced understanding of deep fault structure and controls on seismogenic depth. Friction experiments at elevated temperatures re- Understanding the structure and dynamics of the continental li- vealed a transition from velocity-weakening to velocity-strengthening thosphere and the active faults it contains is of fundamental im- steady state friction with increasing temperature (e.g., Dieterich, 1978; portance. While it is well established that faults are highly localized Ruina, 1983; Tullis and Weeks, 1986; Blanpied et al., 1991; Blanpied within the cold, brittle upper crust, less is known about fault structure et al., 1995). Drawing upon available experimental friction data and in the warmer, and more ductile, lower crust and upper mantle. Classic estimates of continental geotherms, Tse and Rice (1986) demonstrated studies of lithospheric stress profiles were based upon laboratory rock that the primary features of crustal faulting (such as seismogenic depth deformation experiments, spanning both brittle and ductile behavior, and recurrence intervals) could be reproduced in models with ideally and theoretically based thermally activated creep laws (e.g., Byerlee, elastic off-fault response. In their model, the fault continues as a loca- 1978; Goetze and Evans, 1979; Brace and Kohlstedt, 1980; Sibson, lized surface below the seismogenic depth, with tectonic displacement 1982; Sibson, 1984). The onset of creep at elevated temperatures re- accommodated by aseismic sliding at depth. This established the im- duces stress within the lithosphere below the frictional strength, portance of the frictional seismic-aseismic transition. The majority of thereby preventing seismic slip. The predicted seismogenic depth of earthquake modeling studies that followed (e.g., Rice, 1993; Lapusta ∼10 to 20 km broadly matches observed depths of seismicity. In these et al., 2000; Kaneko et al., 2011) have followed Tse and Rice (1986) by classic studies, faults below the seismogenic depth are assumed to neglecting the viscoelastic response of the off-fault material. However, broaden rapidly into wide mylonite zones (e.g., Scholz, 2002). there have been a few studies that examine the role of viscoelasticity on These concepts were refined over the following decades to provide a earthquakes, and these are discussed at the end of this introduction.

* Corresponding author. E-mail address: [email protected] (K.L. Allison). https://doi.org/10.1016/j.tecto.2017.10.021 Received 20 May 2017; Received in revised form 14 October 2017; Accepted 20 October 2017 Available online 02 November 2017 0040-1951/ © 2017 Elsevier B.V. All rights reserved. K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

Studies of exhumed fault zones place constraints on the structure of upper crust and the dry upper mantle, both of which are stronger (e.g., faults, degree of localization, and deformation mechanisms at depth. Chen and Molnar, 1983; Burov and Diament, 1995). In the “crème Exhumed rocks preserve evidence of distributed ductile deformation at brûlée” model, the upper and lower crust overlie a much weaker upper depth in the form of broad mylonite zones. Some zones are relatively mantle (e.g., Jackson, 2002; Burov and Watts, 2006). The “crème narrow, such as the 1–2 km wide mylonite zone from the middle and brûlée” model is supported, in regions such as the Mojave Desert, by lower crust beneath the Alpine Fault (Norris and Cooper, 2003). Others, estimates of rheological structure from transient postseismic deforma- such as the Great Slave Lake shear zone in Canada (Hanmer, 1988), are tion and exhumed xenoliths (summarized in Fig. 3 and discussed in tens of kilometers in width. However, it is possible that this large width more detail below) (e.g., Johnson et al., 2007; Thatcher and Pollitz, characterizes a broad zone of anastomosing faults, rather than a single 2008). Less clear support includes evidence that earthquakes are con- fault (Norris and Cooper, 2003). The dynamics of shear zones at the fined to the upper and lower crust (Maggi et al., 2000), suggesting that base of the seismogenic layer is also complicated. Exhumed rocks show the upper mantle is too weak to support earthquakes, though there is mutual overprinting of mylonites and pseudotachylytes (e.g., Cole evidence that in some regions earthquakes are able to occur in the et al., 2007; Frost et al., 2011; Kirkpatrick and Rowe, 2013), which upper mantle (e.g., Inbal et al., 2016). Additionally, evidence based on argues that the same material can respond in different ways to loading measurements of the effective elastic thickness Te, which is a proxy for across a broad range of strain rates. This evidence also shows that the lithospheric strength (Watts et al., 2013), is conflicting. In some re- brittle-ductile transition takes place gradually over a range of depths. gions, Te is smaller than the seismogenic thickness (Maggi et al., 2000; Seismic imaging studies also place constraints on the structure of Jackson, 2002), which suggests that lithospheric strength resides in the faults and degree of localization at depth. Seismic reflection and re- crust, supporting the “crème brûlée” model. Interpretations of both the fraction studies show that many continental transform faults penetrate confinement of earthquakes to the crust and Te relative to the seismo- the entire crust, persisting as localized features that cut through the genic thickness are debated, however. The apparent absence of earth- Moho (Lemiszki and Brown, 1988; Vauchez and Tommasi, 2003), in- quakes in the upper mantle may result from a strong but frictionally cluding the San Andreas (Henstock et al., 1997; Lemiszki and Brown, stable upper mantle (Jackson, 2002). Additionally, others argue that Te 1988; Zhu, 2000), the San Jacinto (Miller et al., 2014), and the Dead is larger than the seismogenic thickness in some regions (Burov and Sea transform faults (Weber et al., 2004). In contrast, the Alpine Fault Watts, 2006), which suggests that the upper mantle may be strong, and and the Marlborough Fault system, both in New Zealand's South Island, thus that the “jelly sandwich” model is correct. Burov and Watts (2006) do not appear to be highly localized at the Moho. Instead they form also find that only the “jelly sandwich” model can support mountain broad zones of deformation in the lower crust and upper mantle (Klosko ranges over millions of years. It is important to bear in mind that each et al., 1999; Molnar, 1999; Wilson et al., 2004). The width of the de- of these apparently conflicting pieces of evidence measures the strength formation zones in the upper mantle beneath these faults, as inferred of the lithosphere over a different timescale. Postseismic deformation, from seismic anisotropy studies of shear-wave splitting, is quite broad: which generally favors the “crème brûlée” model, measures the

200 km for the Alpine Fault (Duclos et al., 2005; Baldock and Stern, strength of the lithosphere on the timescale of a decade, while Te 2005), 130 km for the San Andreas Fault (Bonnin et al., 2010), and measures the strength on the scale of a million years. We focus on the 20 km for the Dead Sea Fault (Rümpker et al., 2003). The Dead Sea time scale of the postseismic and interseismic periods of the earthquake Fault has accommodated comparatively little displacement, 100 km in cycle, which range from a decade to a few hundred years, and therefore contrast with 850 km for the San Andreas, and this may account for the consider the “crème brûlée” model. Furthermore, there are likely re- relatively narrow zone exhibiting anisotropy (Baldock and Stern, 2005). gional variations in the strength of the lower crust and upper mantle. Anisotropy beneath the North Anatolian Fault does not align with the Transient postseismic deformation in the first few months to years strike of the fault, perhaps indicating that the width of the shear zone is after an earthquake provides short-term constraints on the rheological < 50 km, smaller than can be imaged by shear-wave splitting (Vauchez structure of the crust. The three commonly considered mechanisms are and Tommasi, 2003), or that the fault is too young to have produced frictional afterslip (Perfettini and Avouac, 2004), viscoelastic flow measurable anisotropy (Berk Biryol et al., 2010). This shear-wave (Pollitz et al., 2000; Freed and Bürgmann, 2004; Freed et al., 2007), and splitting occurs at greater depths than we consider in our model, so we poroelastic rebound (Booker, 1974; Jónsson et al., 2003; Fialko, 2004). don’t seek to produce or explain the extent of these deformation zones, Complicating matters, it is possible for multiple mechanisms to be ac- but rather include this discussion to describe the extent to which de- tive after the same event (e.g., Rousset et al., 2012), and, for strike-slip formation broadens with depth. faults, frictional afterslip and viscoelastic flow can produce similar It is evident that considerable questions remain unresolved with deformation signals on Earth's surface (Thatcher, 1983; Hearn, 2003). regard to the frictional velocity-weakening to velocity-strengthening We focus on frictional afterslip and viscoelastic flow, as they also have transition on faults and the transition marking the onset of bulk viscous significant implications for the strength of the lithosphere. In particular, flow in rocks surrounding the fault. In this study, we investigate the viscous flow provides constraints on the lateral and vertical rheological interaction between these two transitions in the continental litho- structure (e.g., Thatcher and Pollitz, 2008; Hearn et al., 2009; Vaghri sphere, with particular focus on the interplay between aseismic fault and Hearn, 2012; Yamasaki et al., 2014; Rollins et al., 2015; Masuti creep and viscous flow across the brittle-ductile transition zone. We et al., 2016). Frictional afterslip reveals information about the frictional simulate friction on a strike-slip fault, capturing the transition from properties of the fault, including estimates of the value (Perfettini and shallow coseismic fault slip to deeper interseismic fault creep. We Avouac, 2004; Barbot et al., 2009) and spatial variation of frictional couple this with off-fault viscoelastic deformation, representative of the properties (Miyazaki et al., 2004). In combination with the postseismic deformation of the lower crust and upper mantle. A diagram of our period, data from the interseismic period can be used to infer rheolo- model is shown in Fig. 1. gical parameters (e.g., Hetland and Hager, 2005) or frictional para- The transitions described previously have implications for stress meters (e.g., Lindsey and Fialko, 2016). Some studies find evidence for profiles within the continental lithosphere. The lithosphere is typically coupled frictional afterslip and viscous flow in the postseismic period divided into three rheologically distinct layers: the upper crust, lower (e.g., Biggs et al., 2009; Johnson et al., 2009; Masuti et al., 2016). crust, and upper mantle. The upper crust is brittle, so its strength is We simulate afterslip and viscoelastic deformation in the context of determined by the strength of the fault, governed by Byerlee's law or earthquake cycle simulations. Our philosophy is to start with the sim- rate-and-state friction. The relative strength of the lower crust and plest case that combines spontaneously nucleating ruptures with off- upper mantle depends upon the composition of those layers, as well as fault viscous flow. We begin with an assumed compositional structure water content, and remains an open question. In the “jelly sandwich” and geotherm, assign model parameters for this structure from la- model, the weak, wet lower crust is sandwiched by the cool, brittle boratory experiments, and our simulations determine the recurrence

233 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

Fig. 1. Increasing temperature, and compositional changes, with depth cause two significant transitions: the frictional transition from velocity-weakening to velocity-strengthening friction on the fault and the transition from elastic to viscoelastic deformation in the bulk (with viscous strain denoted by colors within the sche- matic). Our 2D strike-slip fault model simulta- neously captures both transitions, which in general occur at different depths. Remote tectonic loading, applied by displacing the side boundaries at a constant rate, generates earthquakes on the fault and bulk viscous flow. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) interval, coseismic slip distribution, and partitioning of postseismic and viscosity, geothermal gradient, and degree of strain localization in the interseismic deformation into afterslip and bulk viscous flow. This al- lower crust and upper mantle (e.g., Ave Lallemant et al., 1980; Behr and lows the slip, stress drop, and recurrence interval of each earthquake to Hirth, 2014; Chatzaras et al., 2015), shown in Fig. 3. Additionally, es- develop in a way that is self-consistent with the history of earthquakes timates of the effective viscosity structure of the crust from GPS and and postseismic deformation. To date, earthquake cycle simulation ef- InSAR measurements of postseismic and interseismic deformation from forts have primarily focused on the effects of rate-and-state friction the western US are summarized in Johnson et al. (2007) and Thatcher while representing the off-fault material as elastic (e.g., Tse and Rice, and Pollitz (2008), reproduced in Fig. 3. Also included in the figure are 1986; Rice, 1993; Lapusta et al., 2000; Kaneko et al., 2011),oron results from isostatic adjustment from lake loads over thousands of viscoelastic deformation while kinematically imposing the motion of years. An additional compilation of the results of postseismic inversions the fault (e.g., Savage and Prescott, 1978; Thatcher, 1983; Johnson globally can be found in Wright et al. (2013), who find that in general et al., 2007; Takeuchi and Fialko, 2012). A few codes are capable of the Maxwell viscosity in the lower crust and upper mantle ranges from fully coupling both rate-and-state friction and linear or power-law 1017–7×1019 Pa s. The wide range in viscosity estimates results in part viscoelasticity. However, some are limited to a fault-containing elastic from different assumptions made in each model, with some assuming a layer over a viscoelastic half-space (Kato, 2002; Lambert and Barbot, single viscosity in a linear Maxwell rheology (e.g., Segall, 2002; Bruhat 2016), a set-up which specifies a priori the brittle-ductile transition and et al., 2011), others a linear Burgers rheology (Pollitz, 2003, 2005), and cannot permit material in this region to experience both fault slip and some a nonlinear power-law rheology (Freed and Bürgmann, 2004; viscous flow at different times in the earthquake cycle. Others do not Freed et al., 2006). Models which assume a linear Maxwell rheology simulate multiple earthquake cycles (Barbot and Fialko, 2010; Aagaard cannot account for temporal variations in effective viscosity, such as the et al., 2013). These other codes are used mainly to address different observed increase from 1018 Pa s to 1019 Pa s over the course of 5 years questions than the ones explored in our simulations. after the Hector Mine earthquake (Freed and Bürgmann, 2004). This is Lambert and Barbot (2016) focus on viscous flow transients in the accounted for in a linear Burgers rheology by two different viscosities, upper mantle and their signature in surface displacements. They find and in a power-law rheology through the stress-dependence of the ef- that bulk viscous flow can contribute significantly to postseismic sur- fective viscosity. Additionally, while the models summarized in Fig. 3 face deformation quite early in the postseismic period. Also, while near- assume no spatial variation, some studies have found evidence for a field postseismic surface deformation is dominated by frictional after- localized zone of lower viscosity near the fault (e.g., Yamasaki et al., slip, the contribution from viscous flow can be comparable in the far 2014). field within one year of rupture. Using a similar model geometry, Kato (2002) focuses on the recurrence interval and average slip of each 3. Model earthquake, finding that neither are appreciably affected by flow in the viscoelastic half-space. Shimamoto and Noda (2014) use a unified We consider the two-dimensional antiplane shear problem of a constitutive law to transition from shallow frictional behavior to deeper vertical strike-slip fault. For computational efficiency, we use the quasi- viscous flow (in a localized fault root). Like Kato (2002), they also find dynamic approximation to elastodynamics (quasi-static elasticity with that the behavior of the seismogenic crust is quite similar to that of a the radiation damping approximation, Rice, 1993). We also assume purely frictional model. Additionally, they find that coseismic slip is material properties and tectonic loading are symmetric about the fault, able to penetrate into the brittle-ductile transition region, resulting in a enabling us to model only one half of the domain (y ≥ 0). As shown in region about 5 km wide in which both coseismic fault slip and appre- Fig. 1, the fault is located at y=0 and is parallel to the z-axis, and ciable viscous flow occur. Because the brittle-ductile transition depends displacements are in the x-direction. Note that the fault cuts through on strain rate, the depth extent of coseismic faulting exceeds the brittle- the entire model domain. However, partitioning of the remote tectonic ductile transition depth as estimated from the plate rate. And finally, a loading into fault slip or bulk viscous flow will be determined by the similar modeling study (Beeler et al., submitted) finds that ruptures relative strength of the fault and off-fault material, so the fault will not penetrate 1–2 km below the nominal brittle-ductile transition, but that necessarily slip at depth. less than 10% of the total moment release occurs in this process. The governing equations are

∂σxy ∂σ 2. Lithospheric structure + xz = 0, ∂y ∂z (1) Parameter choices in our study are loosely motivated by the Mojave ⎛ ∂u V ⎞ ∂u V Desert region of Southern California, a well-studied area with a wealth σxy = μ ⎜⎟− γσμ, xz = ⎛ − γ ⎞, ∂y xy ∂z xz (2) of available data. In this region, the top of the upper mantle, deﬁned as ⎝ ⎠ ⎝ ⎠ the depth of the Moho discontinuity, is approximately 30 km deep, as V −−11V γησγησ̇==xy, ̇ xz, shown in Fig. 2. xy xz (3) Insight into the structure of the lithosphere in this region comes in η−−−1/1==AeQRTn τ, τ σ22+ σ , part from exhumed xenoliths, which can provide constraints on the xy xz (4)

234 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

a) Moho depth b) shear modulus 20 0

60 10

40 LA 30 20 μ (GPa) depth (km)

Moho depth (km) Moho 20 30

San Diego 40 40 0 -118 -117 -116 -115 longitude

Fig. 2. (a) Depth of the Moho. Data from the Southern California Earthquake Center's CVMH 15.0 (SCEDC, 2013). The white box indicates the approximate location of the Mojave Desert region. (b) Vertical cross-section along the purple line in (a), showing the shear modulus as a function of depth and the location of the Moho. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

0 force balance on the fault, where stress on the fault, with the radiation- damping approximation, is equal to frictional strength; Eq. (6) is the Upper Crust state evolution; Eq. (7) is the fault slip; and Eq. (8) is the slip velocity. In 10 V these equations, u is the displacement in the x direction, γij are the (engineering) viscous strains, σij are the stress components, μ is the 20 shear modulus, η is the effective viscosity, and T is the temperature. On Lower Crust the fault, τ is the shear stress, σn is effective normal stress, ψ is the state ffi 30 variable, V is slip velocity, f is the friction coe cient, and G is the state fl estimates from Cima evolution equation (e.g., slip law or aging law). In the ow law, the Volcanic Field xenoliths effective viscosity is a function of the rate coefficient A, the activation 40 energy Q, the stress exponent n, and the deviatoric stress τ . The overdot depth (km) estimates from Coyote indicates a time derivative. Lake xenoliths fi 50 As depicted in Fig. 1, the computational domain is nite and there are three other boundary conditions in addition to the fault: estimates from Basin and Range xenoliths Vt 60 u (,,)Lzt= L , y (9) Upper Mantle 2

70 σxz (,0,)yt= 0, (10) 1017 1018 1019 1020 1021 1022 1023 viscosity (Pa s) σxz(,yL z , t )= 0, (11)

Fig. 3. Effective viscosity estimates for the Western US, modified from Johnson et al. where Ly and Lz are the dimensions of the model domain in the y- and z- (2007), Thatcher and Pollitz (2008), and Behr and Hirth (2014). Black lines are viscosity directions, respectively, V L is the tectonic plate velocity. At Earth's estimates from individual relaxation studies, shaded gray rectangles represent the total surface, we use a traction-free boundary condition (Eq. (10)). We also range of viscosity estimates from postseismic studies, and red lines are the average use a traction-free boundary condition at the bottom of the domain (Eq. viscosity estimate for that layer (Thatcher and Pollitz, 2008). The colored boxes indicate viscosities estimated from xenoliths from Southern California: green from the Cima (11)), which permits an arbitrary amount of tectonic displacement to Volcanic Field in the western Mojave (Behr and Hirth, 2014); purple from the Basin and occur, even in an elastic model. It is important to use a large enough Range, assuming the same strain rate as the Cima Volcanic Field xenoliths (Ave Lallemant domain that the simulation results are relatively insensitive to the lo- et al., 1980); yellow from Coyote Lake basalt (Chatzaras et al., 2015). The blue box in- cation of the remote boundaries. Results are more sensitive to the side dicates viscosity estimates for the lower crust based on the Cima Volcanic Field xenolith boundary location, which we place at 120 km from the fault, or 12 fl data and ow laws for wet diabase-anorthite and pure anorthite mixtures in the lower times the seismogenic depth. Results are much less sensitive to the crust (Behr and Hirth, 2014). (For interpretation of the references to color in this figure depth of the bottom boundary, which in our simulations is placed at legend, the reader is referred to the web version of this article.) 40 km depth. with fault boundary conditions 3.1. Frictional parameters

τzt(, )=−= σxy (0,, zt ) ηrad V /2 fψVσ ( , )n, (5) The fault is governed by the force balance between the frictional ψ̇= GψV(, ), (6) strength of the fault, determined by rate-and-state friction, and the shear stress exerted by the viscoelastic solid on the fault within the δzt(, )= 2 u (0,, zt ), (7) context of the radiation-damping approximation. We use the regular- ized form for frictional strength (Rice et al., 2001), ∂δ V (,zt )= , ∂t (8) −1 ⎛ V ψa/ ⎞ f (,ψV )= a sinh ⎜⎟e , ⎝ 2V0 ⎠ (12) where Eq. (1) is the static equilibrium equation; Eq. (2) is Hooke's law; Eqs. (3) and (4) express the power-law viscous ﬂow law; Eq. (5) is the with the aging law for state evolution (Ruina, 1983; Marone, 1998),

235 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

0 eﬀects, the frictional parameters on the fault control the depth of a: Blanpied et al. (1995) earthquake nucleation and arrest. Thus, the earthquakes nucleate at a - b: Blanpied et al (1995) 400 10 km and arrest 4 km below this. a - b: Blanpied et al (1991) a: model parameters 3.2. Remote stress state a - b: model parameters We set the stress state of the system, on which the frictional strength 10 600 (Eq. (5)) depends, assuming that the ratio of fault shear stress to effective normal stress is approximately equal to the reference coeﬃcient

of friction f0. For an optimally oriented strike-slip fault, the principal stresses are related to each other by σ1 > σV > σ3 and 800 2

depth (km) σ13/1σf=++0 f0, where f0 =0.6 (e.g., Sibson, 1974). We assume σ temperature (K) vertical total stress is equal to lithostatic pressure and that V = 20 (σ1 +σ3)/2. This produces the eﬀective normal stress and shear stress on the fault 1000 σσ13+ σσ 13− σnp= − cos 2ϕP− , 22 (14)

σσ13− σxy = sin 2ϕ, 2 (15) 30 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 ϕfP==×tan−−13 , (1000kg m )gz . a, a-b 0 p (16) ff Fig. 4. Rate-and-state laboratory data for a and a−b for wet granite from Blanpied In Eq. (14), we have assumed that the e ective normal stress is equal to et al. (1991, 1995) shown as dots. The a and a−b profiles used in our simulations are the total normal stress minus the pore pressure, assumed in Eq. (16) to shown as solid lines. The geotherm is 30 K/km. be hydrostatic. Near the brittle-ductile transition, both the assumption that pore pressure is hydrostatic and the dependence of effective normal stress on pore pressure may not hold (Beeler et al., 2016), but bV0 ⎛ ()/fψb− V ⎞ G (,ψV )= ⎜⎟e 0 − . we defer investigation of these ideas to future studies. dc ⎝ V0 ⎠ (13)

The key parameters which determine the frictional behavior of the 3.3. Rheological parameters system are the direct effect parameter a, the state evolution parameter Next, we consider the rheology of the off-fault material. We loosely b, and the state evolution distance dc. Laboratory data indicate that dc is on the order of microns (e.g., Dieterich, 1979; Marone and Kilgore, base the depth and composition of each layer for our model on the 1993); however, we use 16 mm to reduce the computational load. If a Mojave Desert region. As we focus on the brittle-ductile transition and surface has a−b > 0, termed velocity-strengthening (VS), then steady- viscoelastic deformation at depth, we neglect the spatial variation in the state frictional resistance to sliding increases as slip velocity increases shear modulus and depth of the upper mantle, instead using constant and stable sliding of the surface is possible. On the other hand, if values given in Table 1. a−b < 0, termed velocity-weakening (VW), then steady-state frictional The composition of the continental lithosphere varies vertically and ff resistance to sliding decreases as slip velocity increases, resulting in the laterally, and involves a mixture of di erent minerals, but the bulk of fi potential for unstable sliding. In an elastic model, earthquakes will the available laboratory data focuses on speci c mineral phases. In nucleate at the transition between velocity-weakening and velocity- particular, rheological parameters for quartz and olivine are relatively strengthening behavior, called the seismogenic depth. Hereafter, we use well understood (e.g., Hirth et al., 2001; Hirth and Kohlstedt, 2003). that term more generally to refer to the depth of coseismic slip. Mixing laws to determine the rheological properties of a composite To assign distributions for a and b to the fault, we use laboratory material are still in development, however, and depend upon the spe- fi data for wet granite gouge from Blanpied et al. (1991, 1995), re- ci c composition and the geometry of each phase (e.g., Tullis et al., produced in Fig. 4. The wet granite data have been used extensively in 1991). As we cannot hope to explore the entire range of possible ma- earthquake cycle simulations before, such as Rice (1993), Lapusta et al. terial parameters, we consider a single model consisting of quartz- (2000), Lapusta and Rice (2003), Kato (2002), and Lindsey and Fialko diorite in the upper and lower crust and wet olivine in the upper (2016). The data predict a shallow transition from velocity-strengthening to velocity-weakening at about 400 K and a deeper transition Table 1 Values used for model parameters. Power-law flow parameters are listed in Table 2. back to velocity-strengthening at around 600 K; however, we neglect the shallow transition in order to focus on the behavior of the system at Symbol Meaning Value depth. Because the fault cuts through the entire domain of the model, the data have been extrapolated below the last data point. This extra- a Direct effect parameter Depth variable, see Fig. 6a b State evolution effect parameter Depth variable, see Fig. 6a polation is linear based the linear dependence of a on temperature, dc State evolution distance 16 mm which results from the Arrhenius activated process that describes creep η Effective viscosity Depth variable, see Fig. 6b at asperity contacts (Rice et al., 2001). Note that the Blanpied et al. ηrad Radiation damping coefficient 9.5 MPa s/m ffi (1995) data would predict a very weak fault when extrapolated to the f0 Reference friction coe cient for Steady 0.6 sliding base of the lower crust, so we follow the Blanpied et al. (1991) data for −3 Pp Hydrostatic pressure (1000 kg m )×gz − − − a b instead. There is recent experimental evidence that Westerly R Gas constant 8.31 J K 1 mol 1 granite can remain velocity-weakening up to 873 K (Mitchell et al., μ Shear modulus 30 GPa − 2016) under certain conditions, but we defer exploration of that pos- ρ Density of rock 3 g cm 3 ff sibility for future studies. σn E ective normal stress Depth variable − 6 −1 V 0 Reference velocity 10 ms Our choices for frictional parameters put the VW-VS transition at − 9 −1 V L Loading velocity 10 ms 10 km depth. In our simulations, despite the inclusion of viscoelastic

236 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

Table 2 3.4. Lithospheric strength Values used for power-law ﬂow parameters.

− − − The choices of frictional and rheological parameters, combined with Material A (MPa n s 1) nQ(kJ mol 1) Source assumed geotherms and constant reference strain rate, produce the li- − Quartz- 1.3×10 3 2.4 219 Hansen and Carter (1982); thospheric strength profiles plotted in Fig. 6c. In the three viscoelastic diorite Freed and Bürgmann simulations considered in this paper, differing only by choice of geo- (2004) therm, the brittle-ductile transition occurs at approximately 15 to Wet olivine 3.6×105 3.5 480 Hirth and Kohlstedt (2003) 20 km depth. To set the geotherm, we use heat flow measurements for the Mojave 3 mantle. Another logical choice for the rheological layers would have Desert region. Surface heat flow in the region is 60–75 mW/m been the traditional “crème brûlée” model, with the upper crust re- (Lachenbruch et al., 1985), resulting in estimates of the geothermal presented by quartzite, the lower crust by feldspar, and the upper gradient ranging from15 to 30 K/km (Williams, 1996). Yang and mantle by olivine. We found, however, that wet quartzite is so weak Forsyth (2008) estimate the Moho temperature to be 973–1098 K based that coseismic slip is entirely confined within a few kilometers of the on seismic attenuation, and the upper end of this range is consistent surface, in contradiction with observed earthquake depths. Ad- with xenoliths from the region (Behr and Hirth, 2014). Based on these ditionally, while quartzite may be representative of the relatively data, and neglecting both heat generation and spatial variations in granitic upper crust, the lower crust is expected to be more mafic crustal conductivity for simplicity, we assume a set of three linear (Hacker et al., 2015; Shinevar et al., 2015). Thus, we instead chose to geotherms: 20 K/km, 25 K/km, and 30 K/km. These three geotherms use quartz-diorite, a material with a much deeper brittle-ductile tran- produce the effective viscosity profiles plotted in Fig. 6b. In order to sition, for the upper and lower crust. Though our choice to use the same study the effect of moving the brittle-ductile transition relative to the composition for all our simulations neglects the rheological complexity earthquake nucleation depth, we use 30 K/km to set the depth-depen- of the crust, the structure is consistent with postseismic data from the dence of the frictional parameters for all simulations. Hector Mine and Landers earthquakes in the Mojave region (Freed and In this study, we assume all viscoelastic deformation happens as a Bürgmann, 2004). Additionally, it enables the model to capture the result of dislocation creep, which is grain-size insensitive and has a essential features on which we wish to focus: the brittle-ductile tran- stress exponent n ≈ 3−5(Rybacki and Dresen, 2000). This assumption sition, and the partitioning between fault motion and bulk viscous flow is well-supported by experimental and microstructural evidence for as a function of depth. quartz (Hirth et al., 2001) and olivine (Hirth and Kohlstedt, 2003), as We smoothly transition the effective viscosity between the crust and summarized in deformation mechanisms maps in Bürgmann and Dresen mantle using a mixing law that applies when the constituent materials (2008). Lower crustal shear zones, however, may have a small enough form layers. Mixing is likely to be the case at the base of the crust, grain size that dislocation creep and diffusion creep could both be ac- where magmatic underplating takes place. The mixing law is derived in tive (Rybacki and Dresen, 2000, 2004). We also neglect the effects of Ji et al. (2003), in their Eqs. (19)–(22), and we now state the equations transient creep, which could reduce the effective viscosity during the for two phases. For a composite of two materials, the power-law postseismic period (Freed et al., 2010; Masuti et al., 2016). parameters are

1 4. Discretization of governing equations nc = , ϕn1//1 + ϕn2 2 (17) We discretize the governing equations using ﬁnite diﬀerences for the spatial discretization and a fully explicit, adaptive Runge-Kutta nϕcc1/ n1 nϕ2/ n2 AAc = ()(),1 A2 (18) time-stepping scheme for the temporal discretization. The spatial discretization, which is described in Appendix A, is provably stable; the proof is based on establishing an energy balance for the semi-discrete ⎛ ϕQ1 1 ϕQ2 2 ⎞ Qcc= n ⎜⎟+ , problem that mimics the mechanical energy balance of the continuum n1 n2 (19) ⎝ ⎠ problem. The fully explicit temporal discretization is possible because the largest stable time step in the interseismic period is limited by the where ϕ1 and ϕ2 are the volume fractions of each phase, shown in minimum Maxwell time, min(T )=min(η/μ), which is the char- Fig. 5. max acteristic timescale for the viscous strains. The smallest Maxwell time in the simulations considered is 20 years, which is not prohibitive relative to the earthquake recurrence interval of approximately 350 years. Be- 0 cause we use the quasi-dynamic approximation, the time-step limitation imposed by rate-and-state friction and radiation damping varies over 1 many orders of magnitude through the coseismic and interseismic 10 2 periods, and thus requires a time step on the order of milliseconds in the coseismic period but is larger than min(Tmax) in the interseismic period. 20 Our algorithm extends that of Erickson and Dunham (2014), which was developed for the linear elastic problem, to the power-law vis-

depth (km) coelastic problem. In addition, we generalize the spatial discretization 30 from uniform grid spacing to variable grid spacing, greatly reducing the computational load, using an approach that is similar to Erickson et al. (2017). We also extend the algorithm to 4th order accuracy in space. 40 0 0.25 0.5 0.75 1 To ensure accuracy of the solution, certain length scales must be resolved by several grid points. From stability analysis of steady fric- volume fraction tional sliding (e.g., Ruina, 1983; Rice, 1983; Rice, 1993; Rice et al., ϕ ϕ Fig. 5. Volume fractions for quartz-diorite ( 1, blue) and olivine ( 2, red). (For inter- 2001), the critical length scale for unstable sliding between elastic half- pretation of the references to color in this ﬁgure legend, the reader is referred to the web spaces with rate-and-state friction is version of this article.)

237 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

Fig. 6. (a) Rate-and-state friction parameters (a) (b) (c) for wet granite (Blanpied et al., 1991, 1995). (b) 0 0 0 Effective viscosity from power-law flow law −14 −1 a 30 K/km γ˙ref =10 s model with quartz-diorite (Hansen and Carter, a-b 25 K/km 1982) in the upper and lower crust and wet 20 K/km olivine (Hirth and Kohlstedt, 2003) in the upper mantle. Three geotherms are plotted: 30 K/km (solid), 25 K/km (dashed), and 20 K/km 10 10 10 (dotted). The gray boxes correspond to effective viscosity estimates from Thatcher and Pollitz (2008). (c) Strength curves: frictional strength (pink) and the viscous strength for each geotherm (blue), showing the brittle-ductile transition occurs at approximately 15 to 20 km depth. 20 20 20 Frictional strength for the Byerlee friction law,

τfσ= 0 n assuming f0 =0.6, is plotted in black for depth (km) reference. The effective viscosities and corresponding strength profiles are plotted for an as- quartz- − − sumed reference strain rate of 10 14 s 1, and diorite do not reflect the variable strain rates which 30 30 30 Byerlee friction occur in our simulations. (For interpretation of fi r+s friction the references to color in this gure legend, the viscous strength: 30 K/km reader is referred to the web version of this ar- wet viscous strength: 25 K/km ticle.) olivine viscous strength: 20 K/km 40 40 40 -0.05 0 0.1 0.2 1018 1020 1022 1024 50 150 250 effective viscosity (Pa s) differential stress (MPa)

μdc 5. Compute rates for the viscous strains and state variable using Eqs. h* = . σbn ()− a (20) (6), (A.20), and (A.21). 6. Update slip δ, the state variable ψ, and the viscous strains γV and γV , Numerical simulations for the aging law have found that an additional xy xz e.g., δn +1 =δn +V nΔt. critical length scale, characterizing the size of the region of rapid strength degradation behind the tip of a propagating rupture, is Earthquake cycle simulations like this must be spun-up, meaning μdc that it takes some (potentially large) number of earthquakes for each Lb = σbn (21) simulation to settle into a limit cycle. While there is no proof that each simulation must settle into a limit cycle, in our experience all power- (Dieterich, 1992; Ampuero and Rubin, 2008). This scale is more chal- law viscoelastic simulations with this value of state evolution distance lenging to resolve. dc seem to do so. We have found that linear elastic simulations take very little time to spin-up, whereas viscoelastic simulations require thou- 4.1. Temporal discretization sands of earthquakes to spin-up if the viscous strains are initialized at zero. This spin-up time scales with the Maxwell time for each simula- The variables integrated in time are slip on the fault δ, the state tion, and reflects the time needed for bulk viscous flow to communicate variable ψ, and the viscous strains γV and γV . The temporal integration xy xz the stress state at the fault to the surrounding material. To reduce the is performed using a third-order accurate, adaptive Runge-Kutta time- computational expense of model spin-up, we have developed a multi- stepping algorithm (Hairer et al., 1993) in order to capture the range of step process: time scales across the earthquake cycle. It is outlined here as if for forward Euler for simplicity. To integrate from step n (time tn) to step n 1. This step determines the displacement boundary condition on the +1 (time tn+1 =tn +Δt) over time step Δt: fault that is consistent with the predicted stress profile on the fault n n n n (as shown in Fig. 6c). We do not integrate in time, and enforce 1. Set boundary conditions: u(Ly,z,t )=V Lt /2, u(0,z,t )=δ /2. traction on the fault rather than displacement, setting the traction to 2. Solve the static equilibrium equation Eq. (1) for displacement un: that of the predicted stress profile for a reference strain rate of n Vn,,Vn − 14 − 1 AbCDuγγ=+xy + xz , (22) 10 s . The remote boundary displacement may be set to any value. The viscous strains are set to zero. We solve the static equi- where A, C, and D are matrices set by the spatial discretization Eq. librium equation (Eq. (1)) for displacement and evaluate it on the (A.15) and b is a vector containing boundary data. fault. This provides an initial guess for the fault displacement 3. Use displacement un to compute the quasi-static shear stress on the boundary condition that is consistent with a plausible lithospheric fault: strength profile. n 2. This step provides a fault shear stress profile that is more consistent n ⎛ ∂u Vn, ⎞ σxy = μ ⎜⎟− γxy , with the rate-and-state friction law, without assuming a reference ⎝ ∂y ⎠ y=0 slip velocity and strain rate. We switch to the full problem, with where the discretized form is given in Eq. (A.13). rate-and-state friction on the fault and nonzero viscous strains in the 4. Solve the nonlinear force balance equation on the fault (Eq. (5)) for bulk, and use the displacement from step 1 to set the initial fault slip velocity V n: displacement. We set the initial viscous strains to zero everywhere in the domain, and initialize the state variable as f . We integrate in σ fψ(,nn V )=− σn η Vn. 0 n xy rad time until the shear stress on the fault settles into a limit cycle. Note that the stress on the fault reaches a limit cycle far sooner than does

238 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

the off-fault stress, since the fault stress does not require viscous 365 years for the 30 K/km simulation, 375 years for the 25 K/km si- flow to accumulate. In our experience, the fault shear stress stabi- mulation, and 366 years for the 20 K/km simulation. This suggests that lizes after of order 10 earthquakes. the loading of the seismogenic crust is effectively the same, regardless 3. This step allows bulk viscous flow to propagate the stress state at the of the deformation mechanism at depth, an idea which we discuss fault to the off-fault material without requiring the computational further below. load necessary to model earthquakes. We enforce traction on the While plots of cumulative fault slip provide some insight into the fault using a representative interseismic shear stress profile from depth-dependence of bulk viscous flow, they do not show its spatial V V ff step 2 as the fault boundary condition, and corresponding γxy and γxz distribution. Both components of the o -fault viscous strain that accu- V as initial conditions. We integrate in time until the elastic strain at mulate over a single earthquake are plotted in Fig. 8. Viscous strain γxy, each depth reaches a steady state value (see Eq. (23) and sur- the component that accommodates tectonic displacement, can be ex- rounding discussion). This eliminates the need to take very small amined to determine the width of the shear zone beneath the fault. The V time steps during the coseismic period, thus greatly decreasing the peak of γxy occurs at the depth of least interseismic fault creep, which number of time steps needed to spin-up the simulation. It is possible becomes shallower with increasing geotherm. Also, the width of the to further speed up this step by using a larger grid spacing, as the shear zone is on the order of a few kilometers at this depth, and critical grid spacing for rate-and-state friction does not need to be broadens significantly below this. The shear zone is narrowest for the resolved, followed by an interpolation onto the finer grid. warmest geotherm, though the 25 K/km and 30 K/km simulations are 4. We use the final boundary displacements and viscous strains from quite similar. The 20 K/km simulation, however, produces a much step 3 as the initial conditions for a full simulation with rate-and- broader shear zone. This general dependence of shear zone width on state friction on the fault. For the state variable, we use the final geotherm is in agreement with the analytical calculations of Moore and value from step 2. It still takes a few earthquakes to reach the de- Parsons (2015), who find that the primary control on shear zone width V sired limit cycle, but not thousands. is the geotherm. γxz relates to basal tractions on the lithosphere, which V we also discuss in Section 5.3. Where γxz is negative, the upper crust 5. Results drags the lower crust, and the lower crust in turn drags the upper V mantle. The largest values of γxz occur close to the fault: within 5 km of In this section, we summarize the results of the three viscoelastic the fault for the 20 K/km simulation, and within 15 km for the 25 and simulations and a reference elastic simulation. We discuss features of 30 K/km simulations. the earthquake cycle, such as the recurrence interval and nucleation Strikingly, in many respects the coseismic phase of each viscoelastic depth, which contain information about the ways in which bulk viscous simulation is quite similar to that of the linear elastic simulation. In flow at depth loads the seismogenic zone. We also examine the spatial every case, the earthquakes nucleate at the same depth with approxi- distribution and degree of localization of viscous strain. We begin by mately the same recurrence interval. Additionally, the down-dip limit presenting slip histories and viscous strain distributions in Section 5.1. of coseismic slip in each simulation is 14 km depth, which means that it The variation in viscous strain rate in the postseismic and interseismic is controlled by the transition in frictional parameters. These simila- periods is addressed in Section 5.2, as is the relative partitioning of fault rities in recurrence interval and coseismic slip distribution between slip into coseismic slip and frictional afterslip. The stress in the litho- elastic and viscoelastic simulations were also observed by Kato (2002). sphere for each model is discussed in Section 5.3. We also discuss the The similar behavior can be explained by the fact that in each simu- sign and magnitude of basal tractions within the lithosphere, which lation no appreciable viscous strain occurs above 14 km depth, and addresses the question of whether the upper crust drags the lower crust therefore the same amount of interseismic fault creep occurs at – and upper mantle or vice versa. Then, we summarize the spatial and 10 14 km depth. This creep likely provides much of the loading to the temporal variation of effective viscosity, which also relates to the seismogenic zone and the upper crust. Thus, the shear stress on the strength of the lithosphere. Finally, we discuss the means by which fault, and therefore the recurrence interval, are approximately the observations of surface deformation might be able to differentiate the same. This apparent similarity is limited, however, and does not hold fi models in Section 5.5. for the evolution of the stress elds below the seismogenic zone with time, nor for the surface deformation. 5.1. Fault slip and viscous strain 5.2. Partitioning of tectonic loading into fault slip, elastic, and viscous Cumulative slip on the fault over a sequence of earthquakes span- strains ning about 2000 years is plotted in Fig. 7. In (a), the linear elastic simulation, the fault creeps steadily at depth at the tectonic loading rate. Figs. 7 and 8 demonstrate that the relative partitioning of the tec- This creep causes a stress concentration to build up at the base of the tonic loading into fault slip and bulk viscous flow varies with depth. seismogenic zone, and when the stress concentration is large enough, This is shown in more detail in Figs. 9 and 10 for the simulation with a an earthquake nucleates at that depth. In this case, the earthquakes geotherm of 30 K/km. At each depth, the contributions of fault slip, occur every 366 years and produce about 10 m of slip. This slip is quite viscous strain, and elastic strain must add up to the tectonic loading large due to the absence of heterogeneity as well as our assumption of displacement, meaning that at all times and depths hydrostatic pore pressure, which results in a large stress drop for each L Ly y V E earthquake. At the surface, the fault also creeps a small amount in the VL tδ=+22 γdy + γdy, ∫∫00xy xy (23) interseismic period as a result of the high h* (from low effective stress) – E in this region. In contrast, in the viscoelastic simulations (b) (d) fault where γxy is the elastic strain. The left-hand side is the tectonic dis- creep in the lower crust and upper mantle lags behind fault slip in the placement, and the second and third terms on the right-hand side seismogenic zone because the off-fault material at depth is weak en- correspond to the contributions of viscous and elastic strain, respec- ough that it inhibits interseismic fault creep. Instead, bulk viscous flow tively. Because the off-fault stresses return to the same level at the start accommodates much of the tectonic loading, preventing the fault from of each earthquake, the integrated elastic strain cannot grow or decay creeping at the loading velocity. As the geotherm increases, the pro- over multiple earthquake cycles, but instead must oscillate about a portion of the loading that is partitioned into bulk viscous flow in- constant value. Only fault slip and integrated viscous strain may increases, such that in (d), with a geotherm of 30 K/km, the fault does not crease with time to keep up with the tectonic loading displacement. slip below 16 km depth. For the power-law simulations, the recurrence This oscillation in elastic strain is similar to the kinematically consistent interval is effectively the same as for the linear elastic simulation: models of Devries and Meade (2016).

239 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

(a) linear elastic earthquakes (b) 20 K/km

interseismic creep

bulk viscous flow prevents fault slip

Fig. 7. Cumulative slip on the fault over a sequence of earthquakes for (a) an elastic simulation, and (b)–(d) power-law viscoelastic simulations. Red contours are plotted every second during an earthquake, defined as when the maximum slip velocity exceeds 1 mm/s, and blue contours are plotted every 10 years during the interseismic period. Over the range of geotherms modeled in (b)–(d), the dominant deformation mechanism in the lower crust changes, from fault creep in the 20 K/km simulation to bulk viscous flow in the 30 K/km simulation. Despite this change, each simulation predicts similar nucleation depths, recurrence intervals, and coseismic slip amounts. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

The upper crust is effectively elastic, so at the surface of the Earth, domain does not extend deep enough to capture this. This figure is plotted in Fig. 9, there is negligible viscous strain. Tectonic loading representative of the other two viscoelastic simulations as well, though causes the elastic strain to build during the interseismic periods, and for them the brittle-ductile transition zone is deeper and bigger. In the then slip on the fault during each earthquake releases this elastic strain. 25 K/km simulation it occurs at 16–20 km depth. The 20 K/km simu- A small amount of interseismic creep occurs at the surface as a result of lation produces significant fault creep to the bottom of the computa- the low normal stress, so during the interseismic period the blue line is tional domain, so for this geotherm the purely ductile regime is not not perfectly horizontal and the elastic strain rate decreases slightly modeled. over the interseismic period. At the end of each cycle, integrated elastic fl strain returns to the same position, re ecting the fact that the simula- 5.3. Temporal evolution of stress tion is in a limit cycle. This figure, except for the shallow interseismic fault creep, is representative of the entire seismogenic zone, which In this section, we discuss the spatial and temporal evolution of li- extends down to 13 km depth for this simulation. thospheric stress components σxy (the shear stress acting on vertical Deeper, in the brittle-ductile transition zone ranging from 14 to 17 planes parallel to the fault) and σxz (the shear stress on horizontal km depth for the 30 K/km simulation, plotted in Fig. 10a–c, fault slip planes, with negative σxz indicating that material closer to Earth's sur- fi and integrated viscous strain both accommodate a signi cant portion of face is dragging the material below it). First, we discuss the predictions the loading. At the down-dip limit of coseismic slip, plotted in Fig. 10a, our simulations make of the stress within the lithosphere as a function both coseismic slip and a small amount of viscous strain occur, de- of depth. Then, we discuss the temporal evolution of σxy throughout the monstrating that the earthquake is able to penetrate a small distance interseismic period, as this has implications regarding the trigging of into the brittle-ductile transition zone. Frictional afterslip and inter- earthquakes via viscoelastic stress transfer. Finally, we discuss the im- seismic fault creep occur at this depth as well, and together they ac- plications of the results for σxz for basal tractions. commodate most of the tectonic loading displacement. Surprisingly, at The spatial distribution of σxy at the start of the interseismic period these depths, viscous strain accumulates steadily throughout the is plotted in the first row of Fig. 11. The linear elastic simulation, earthquake cycle, rather than accumulating at a greater rate in the Fig. 11a, has a radically different stress field than the power-law si- postseismic period. mulations. Far from the fault, stress becomes uniform with depth, a ff Adi erent behavior occurs in the purely ductile regime well below result of the remote boundary condition of uniform displacement and the brittle-ductile transition. The purely ductile regime starts at about depth-independent elastic material response. In contrast, the power-law 17 km depth and extends to the bottom of the model domain. As seen in simulations (Fig. 11g, m, and s) produce near-fault stress profiles that – Fig. 10d f, all of the tectonic loading is accommodated by viscous strain resemble the lithospheric strength profile plotted in Fig. 6c. Far from and no fault slip occurs. Viscous strain does not accumulate at a uni- the fault, the distribution changes and the prediction is that lithospheric fi form rate, however. More viscous strain accumulates during the rst strength will reside in the upper crust, consistent with the fact that all of – half of the interseismic period than the last half. Comparing Fig. 10c e, these simulations are “crème brûlée” models. it is apparent that the viscous strain rate transitions from uniform in An alternative representation of the spatial distribution for σxy for time in the brittle-ductile transition zone to variable down to the the 30 K/km simulation is shown in Fig. 12, which shows series of ffi bottom of the computational domain (at 40 km depth). At su ciently vertical transects through Fig. 11s at selected times. Within a few great depth, with spatially uniform remote loading, the viscous strain kilometers of the fault, plotted in black, the stress profile in the lower − − rate must become uniform with time, though our computational crust is well-approximated by a reference strain rate of 10 12 s 1.In

240 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

Fig. 8. Viscous strain accumulated through one earthquake cycle. Also contoured in (a)–(c) are various measures of the width of the viscous shear zone, defined as the distance from the fault at each depth at which V the cumulative integral of γxy reaches 0.5, 0.75, and 0.9 times its total value. For each V V geotherm, the maxima for γxy and γxz occur at the same depth, which reflects the depth of least interseismic fault creep. Below the maxima, each viscous strain component becomes more diffuse with depth. Note the change in color scale between (a)–(c) and (d)–(f).

the far ﬁeld, plotted in red, the stress proﬁle is better approximated by a simulation. In all four simulations, the shear stress evolves very simi- − − much lower reference strain rate of 2.5×10 17 s 1, for which the larly in the brittle upper crust. lower crust is predicted to be quite weak. These plots highlight the fact Returning to Fig. 11, the viscoelastic simulations also predict un- that the average strain rate is orders of magnitude higher near the fault realistically high values of σxy in the upper crust (200–250 MPa). To than away from it. Additionally, Fig. 13 shows the shear stress on the explain these stresses, we calculate a global force balance on the gray fault at selected times throughout the interseismic period for each rectangular region shown in Fig. 14, which extends from the fault to the

40 Fig. 9. Tectonic loading displacement (gray) is parti- z = 0 km tioned into fault slip (blue), integrated viscous strain (black), and integrated elastic strain (red) at the surface 30 tectonic loading for the 30 K/km simulation. The change in each quantity integrated elastic strain has been plotted, not the absolute value. This is in the fault slip brittle regime, which extends down to 13 km for this integrated viscous strain 20 geotherm, so no viscous strain accumulates. At the end of shallow interseismic interseismic strain build up each cycle, elastic strain returns to the same level, re- fault creep ﬂ

(meters) ecting the fact that the simulation is in a limit cycle with 10 periodic earthquakes. (For interpretation of the references coseismic release to color in this ﬁgure legend, the reader is referred to the of elastic strain web version of this article.) 0 -5 0 500 1000 1500 time (years)

241 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

Brittle-Ductile Transition Zone Ductile Regime

40 40 (a) z = 14 km (d) z = 20 km 30 30 afterslip tectonic loading 20 20 integrated elastic strain fault slip (meters) 10 coseismic slip 10 integrated viscous strain 0 0 -5 -5 0 500 1000 1500 0 500 1000 1500

40 40 (b) z = 15 km (e) z = 30 km 30 30

20 20

(meters) 10 10

0 0 -5 -5 0 500 1000 1500 0 500 1000 1500

40 40 (c) z = 16 km (f) z = 40 km 30 30

20 20

(meters) 10 10

0 0 -5 -5 0 500 1000 1500 0 500 1000 1500 time (years) time (years)

Fig. 10. Partitioning of tectonic loading displacement (gray) into fault slip (blue), integrated viscous strain (black), and integrated elastic strain (red) for the 30 K/km simulation. The change in each quantity has been plotted, not the absolute value. In the brittle-ductile transition zone, beginning at the down-dip limit of coseismic slip (14 km), viscous strain accumulates at a constant rate throughout each cycle; coseismic and postseismic fault slip are clearly visible. In the ductile regime, in which no fault slip occurs, viscous strain accumulates more quickly early in each interseismic period, though this lessens with depth. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) remote side boundary and from the top free surface to some arbitrary 1990 GN/m. Therefore the tectonic driving force is quite high depth. The depth of the bottom boundary of the rectangle is varied. The (2020 GN/m). However, negligible load can be carried by the lower tectonic driving force per unit distance along strike (i.e., integrated crust and mantle far from the fault, which are quite weak because of the stress along a vertical cross-section at the remote boundary) is balanced low strain rate. Hence stresses in the upper crust away from the fault by the sum of integrated basal traction (along the bottom of the rec- become quite high. tangular region) and integrated stress along a vertical cross-section In the real Earth, such high stresses would not occur. Instead, they along the fault and its deep extension. To meet the imposed traction- might be relaxed by slip on additional, subparallel strike-slip faults, free boundary conditions on the top and bottom of the computational each of which might elevate the basal traction at the base of the upper domain, the integrated basal traction must vanish when the bottom of crust just below it and therefore successively decrease the load borne by the rectangular region coincides with either the top or bottom the remote upper crust. While these basal tractions are not too large, the boundary. However, the contribution from basal traction is nonzero, net effect of the basal tractions from several such fault roots could and determined by the rheology and deformation style, when the eventually contribute to the force balance. However, with multiple bottom of the rectangular region is placed elsewhere within the com- faults, the slip rates on each fault would be lower (for the same tectonic putational domain. The basal traction at the boundary between the loading rate), and therefore the net effect of all fault roots on the basal lower crust and upper mantle is relatively low. For example, for the tractions might be similar to that of a single fault model. Nonetheless, 30 K/km simulation, the integrated basal traction is 30 GN/m at 30 km. there are many places where plate boundary strike-slip faults do not The total resistive force acting on the fault cross-section, however, is exist in isolation, but instead occur as a set of subparallel structures, high because the region below the fault has a high strain rate, and such as the Marlborough Fault system in New Zealand (e.g., Wilson hence high stress. For the 30 K/km case, the force on the fault is et al., 2004) and the San Andreas-San Jacinto-Elsinore sytem in

242 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

Linear Elastic 20 K/km 25 K/km 30 K/km (a) 0 years (g) 0 years (m) 0 years (s) 0 years (MPa)

(b) 1 year (h) 1 year (n) 1 year (t) 0 year (MPa)

(d) 10 years (j) 10 years (p) 10 years (v) 10 years (MPa)

(e) 50 years (k) 50 years (q) 50 years (w) 50 years (MPa)

(f) 100 years (l) 100 years (r) 100 years (x) 100 years (MPa)

distance from fault (km) distance from fault (km) distance from fault (km) distance from fault (km)

Fig. 11. Comparison between vertical shear stress σxy (color scale and contours) at selected times in the interseismic period for the linear elastic and power-law simulations. From the left, the first column shows the results for a linear elastic simulation, the second shows the results for the viscoelastic simulation with the geotherm of 20 K/km, the third 25 K/km, and the fourth 30 K/km. The first row shows σxy at the start of the interseismic period, using the colorbar shown on the far right, and all remaining rows show the perturbation in σxy from this value. The title for each plot states the time since the last earthquake. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Southern California (e.g., Lundgren et al., 2009), and our model sug- 2009; Brantut and Platt, 2016), and grain size reduction, foliation, and gests a possible reason for these structures. Alternatively, off-fault de- thermo-mechanical feedback (viscosity reduction from shear heating) viatoric stress in the upper crust might be limited by the occurrence of in the ductile root of the fault (e.g., White et al., 1980; Montési and distributed plastic deformation (e.g., Bird, 2009; Shelef and Oskin, Zuber, 2002; Bürgmann and Dresen, 2008; Platt and Behr, 2011; 2010; Herbert et al., 2014; Erickson et al., 2017), which in the brittle Takeuchi and Fialko, 2012; Montési, 2013). crust might manifest as multiple smaller faults (Evans et al., 2016). The remaining rows of Fig. 11 show the change in σxy relative to the Another possible way to reduce the stress in the upper crust, without start of the interseismic period. This is equivalent to plotting changes in invoking additional faults or plasticity, is to increase basal tractions. Coulomb stress for a vertical strike-slip fault. In the linear elastic si- While we have not seen evidence for this in our simulations, it might be mulation, the change in stress over the first 10 years is that of a crack possible that some rheologies, such as a stronger, more mafic lower located between 12 and 20 km, the depth of frictional afterslip. After crust, would produce substantial basal tractions. Finally, various pro- this, the stress state transitions to that of a screw dislocation, resulting cesses could weaken the fault and its deep extension. These include from the fault's transition to locked in the upper crust and creeping in pore pressures in excess of hydrostatic (e.g., Rice, 1992) as well as the lower crust and upper mantle. Despite the dramatic differences in dynamic weakening mechanisms such as thermal pressurization and absolute value of shear stress for the linear elastic simulation and the

ﬂash heating within the seismogenic zone (e.g., Rice, 2006; Noda et al., 20 K/km viscoelastic simulation, the changes in σxy are nearly identical

243 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

at fault at remote boundary

from shallow interseismic fault creep 0 0 0

20 20 20 nucleation of left by previous stress concentration new earthquake earthquake caused by fault creep depth (km) depth (km) depth (km)

(a) 0 years (b) 10 years (c) 366 years 40 40 40 50 150 250 350 50 150 250 350 50 150 250 350 (MPa) (MPa) (MPa) xy xy xy

Fig. 12. Vertical transects of σxy for 30 K/km simulation at selected times, matching Fig. 11s, starting at the fault (black) and moving to the remote boundary 120 km away from the fault (red). The time since the end of the previous earthquake is listed in the bottom right of each panel: (a) the start of the postseismic period, (b) early in the interseismic period, and (c) at the − − − − end of the interseismic period and at the start of the next earthquake. Also shown are the predicted viscous strengths for reference strain rates of 2.5×10 17 s 1 (dotted), 10 14 s 1 − − − − (dashed), and 10 12 s 1 (solid). On the fault, the stress profile reflects viscous strength profile shown in Fig. 6c with a reference strain rate of 10 12 s 1, while the remote stress is best − − described with a reference strain rate of 2.5×10 17 s 1. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

0 stress elastic 20 K/km from creep from near 5 concentration surface creep from earthquake 25 K/km postseismic 30 K/km creep 10 nucleation 15 }

20 depth (km) 25

35 (a) end of (e) start of next earthquake (b) 1 month (c) 10 years (d) 300 years earthquake 40 0 100 200 0 100 200 0 100 200 0 100 200 0 100 200 τ (MPa) τ (MPa) τ (MPa) τ (MPa) τ (MPa)

Fig. 13. Shear stress on the fault at various times throughout the interseismic period: (a) and (b) show the fate of the stress concentration left behind by the earthquake; (c) postseismic creep causes the development of a stress concentration just below 10 km depth; (d) the stress concentration grows and moves up the fault; and (e) the nucleation of the next earthquake. Despite the diﬀerences in stress level in the lower crust and upper mantle, the shear stress is very similar in the brittle upper crust in each simulation.

throughout the interseismic period, as very little bulk viscous flow oc- The stress component σxz throughout the interseismic period is curs in this viscoelastic simulation. For the warmer two viscoelastic shown in Fig. 15 for each power-law simulation. As is the case for σxy, simulations, the stress distributions are quite similar for the first σxz is largest in the portion of the crust that accommodates significant 5 years, as very little viscous flow occurs over this short time frame. brittle deformation, but tends to small values beyond ∼10 km from the

After this, bulk viscous flow at depth produces a very different stress fault. The sign of σxz is of particular significance: negative σxz means state below 20 km. These models also predict a larger shallow pertur- that shallower material is dragging the deeper material. Thus, for the bation in shear stress at each time. The implication is that the increased simulation with a geotherm of 30 K/km (Fig. 15g–i), the upper crust is bulk viscous flow in the warmest viscoelastic model transmits the effect dragging the lower crust and mantle. For the simulation with a geo- of the earthquake farther from the fault than does the coolest viscoe- therm of 20 K/km (Fig. 15a–c), however, a change in sign occurs. lastic model. Within 5 km of the fault, the upper mantle is dragging the lower crust,

244 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

(a) linear elastic (b) 20 K/km (c) 25 K/km (d) 30 K/km resistive basal traction 0 0 0 0 driving tectonic loading 5 5 5 5 resistive frictional force/length 10 10 10 10

15 15 15 15

20 20 20 20 line integrals used to depth (km) 25 25 25 25 determine the magnitude of each driving/resisting 30 30 30 30 force. model domain 35 35 35 35

40 40 40 40 -1 0 2 4 -1 0 2 4 -1 0 2 4 -1 0 2 4 force/length force/length force/length force/length (1012 N/m) (1012 N/m) (1012 N/m) (1012 N/m)

Fig. 14. Balance between remote tectonic driving force (red), resistive frictional force (yellow), and integrated basal traction (blue), all per unit distance along strike, for the gray shaded rectangular region. The force balance is calculated for variable locations of the lower boundary of the rectangle, labeled “depth” in (a)–(d). In the viscoelastic simulations, the lower crust and upper mantle are quite weak far from the fault, causing the upper crust to bear a substantial part of the load and resulting in unrealistically high stresses. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

20 K/km 25 K/km 30 K/km (a) 0 years (d) 0 years (g) 0 years depth (km)

(b) 10 years (e) 10 years (h) 10 years (MPa) depth (km)

Fig. 15. Horizontal shear stress σxz (shown in both color scale and contours) at selected times throughout the interseismic period, with titles indicating the time since the last earthquake.

Each column pertains to a different geotherm: (a)–(c) 20 K/km, (d)–(f) 25 K/km, and (g)–(i) 30 K/km. Past 20 years, the stress field changes relatively little. Negative σxz indicates that the material above is dragging the material below; that is, in (d)–(i) the upper crust is dragging the lower crust and mantle. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

245 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

20 K/km 25 K/km 30 K/km (a) 0 years (c) 0 years (e) 0 years

(b) 366 years (d) 366 years (f) 366 years

Fig. 16. Effective viscosity at the start (first row) and end (second row) of the interseismic period. Effective viscosity is lowest around the fault, especially in the lower crust beneath the seismogenic zone. While the effective viscosity is almost constant in time for the 20 K/km geotherm, it exhibits substantial temporal variation for the higher geotherms. while at greater distances the reverse is occurring. Overall, these results viscosity is not captured in predictions assuming a single reference mean that, for a broad range of geotherms and effective viscosities, the strain rates. In particular, for the 30 K/km simulation, the effective crust loads the mantle. Our results are consistent with the predictions of viscosity is close to constant with depth in the lower crust and upper Lachenbruch and Sass (1992): a weak lower crust must result in low mantle during much of the interseismic period. basal drag. While the simulations presented here encompass a small portion of rheological parameter space, they nevertheless span the range of ef- 5.4. Effective viscosity fective viscosity estimates for the Western US from deformation studies (compare estimates for the Western US, plotted in Fig. 3, with the ff The power of modeling the power-law rheology is that the viscosity modeled e ective viscosities plotted in Fig. 17). The broad range of the structure is not imposed. Instead, a spatially and temporally variable estimates could suggest that the behavior of the lower crust is in- ff viscosity structure develops that is consistent with the stress state. sensitive to the e ective viscosity; however, our simulations make very ff Fig. 16 shows the spatial distribution for effective viscosity at the start di erent predictions for how deformation is accommodated within the and end of an interseismic period for each simulation. The vertical lower crust and upper mantle, and the associated stress state in the ff structure is set predominantly by the geotherm, so in the upper crust lithosphere. Thus, the simulations demonstrate that the e ective visc- the effective viscosity is very high, meaning that the upper crust is ef- osity estimates imply a great deal of uncertainty in the predominant fectively elastic. Additionally, a zone of relatively low effective visc- deformation mechanism of the lower crust. osity has developed within 20 km of the fault at depth. As was the case V for the viscous strain γxy, this zone broadens with depth. 5.5. Postseismic deformation As demonstrated by Fig. 16, the effective viscosity varies throughout the interseismic period. Fig. 17 shows the extent of this variation for a In this section, we examine the predicted surface deformation vertical transect 1 km from the fault during one interseismic period. In throughout the postseismic and interseismic periods. We also discuss the 20 K/km simulation, the effective viscosity is approximately con- relative contributions of frictional afterslip and bulk viscous flow to the stant throughout the interseismic period, whereas in the 25 K/km si- total surface deformation. While it can be challenging to disentangle mulation it varies by a factor of two in the lower crust, and in the 30 K/ the effects of each deformation process in inversions of crustal de- km simulation it varies by more than an order of magnitude. This factor formation data, this is straightforward in our forward-modeling con- of ten increase in effective viscosity through the postseismic period in text. the warmest simulation is consistent with estimates from postseismic Fig. 18a shows the displacement 20 km from the fault on Earth's surface deformation data for the Hector Mine earthquake (Freed and surface over three earthquake cycles. Both the total displacement and Bürgmann, 2004) and for the Denali Fault earthquake (Freed et al., the contribution from fault slip are plotted (the difference being the 2006). Most of this variation occurs in the first decade of the inter- contribution from bulk viscous flow). While in the 20 and 25 K/km seismic period. Additionally, the effective viscosity profiles predicted simulations, the station moves at an approximately uniform rate over − − − − by constant reference strain rates ranging from 10 16 s 1 to 10 12 s 1 much of the interseismic period, in the 30 K/km simulation the station consistently overestimate the viscosity of the lower crust, especially for experiences more displacement in the first third of the interseismic the two coolest geotherms. Furthermore, the shape of the effective period than in the last third. Additionally, for the 25 and 30 K/km

246 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

(a) 20 K/km (b) 25 K/km (c) 30 K/km

15 15 15

20 20 20

25 25 25

depth (km) 30 depth (km) 30 depth (km) 30

35 35 35

40 40 40 1018 1020 1022 1024 1018 1020 1022 1024 1018 1020 1022 1024 effective viscosity (Pa s) effective viscosity (Pa s) effective viscosity (Pa s)

Fig. 17. Effective viscosity on a vertical transect 1 km from the fault throughout the interseismic period. The solid blue line indicates the start of the interseismic period, the dashed blue − − − − line indicates the end, and the red lines are plotted every 5 years in between. The gray lines show the effective viscosity assuming a constant strain rate of 10 12 s 1 (solid), 10 14 s 1 − − (dashed), and 10 16 s 1 (dotted). Effective viscosity is essentially constant in time for the coolest simulation, and varies considerably for the warmest. In all three simulations, the constant reference strain rate prediction fails to capture the variability in effective viscosity with depth. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) simulations, bulk viscous flow is responsible for most of the station's shorter time frame and closer to the fault than bulk viscous flow. For motion. In contrast, in the 20 K/km simulation, fault creep at depth each simulation, the contribution of fault creep is greatest within 25 km contributes about half of the station's motion. of the fault, while bulk viscous flow is significant even 50 km from the Fig. 18b and c show the displacement in the postseismic period. fault. In addition, the fact that the dashed lines are nearly identical at Over this time frame, afterslip comprises almost all of total displace- 25%T and 75%T means that the fault is creeping at a steady rate by ment for the 20 K/km simulation. In the 25 and 30 K/km simulations, 25%T (T is the recurrence interval). In contrast, the solid lines are not afterslip comprises most of the station motion in the first few months, identical for the 25 K/km and 30 K/km simulations, meaning that bulk with bulk viscous flow becoming increasingly important one year after viscous flow is not occurring at a steady rate. the earthquake. Over most of the first decade of the postseismic period, It may be possible to discriminate between these models using the 30 K/km simulation predicts the most station displacement and the postseismic data spanning years after a large earthquake. For the 20 K/ 20 K/km simulation predicts the least. In the first two months, however, km simulation, most of the station motion can be produced by a model the 25 K/km simulation actually predicts the most station motion. This for frictional afterslip. Also, the component resulting from bulk viscous results from the fact that in this time frame more afterslip occurs in the flow should be well fit with a single effective viscosity that applies both 25 K/km simulation than in the 30 K/km simulation. After two months, to the interseismic data prior to the earthquake and to the entire increased bulk viscous flow causes the 30 K/km simulation to pass the postseismic period. In contrast, the 30 K/km simulation would require a 25 K/km simulation. large contribution from bulk viscous flow, particularly after the first Fault creep at depth and bulk viscous flow contribute to station 6 months. If using a linear rheology, at least two effective viscosities motion for different time frames and at different distances from the would be needed to fit the data. fault, as illustrated in Fig. 19. In general, fault creep is significant over a

(a) (b) 14 0.2 30 K/km: total 12 30 K/km: fault slip 25 K/km: total 0.1 10 25 K/km: fault slip surface

20 K/km: total displacement (m) 0 8 20 K/km: fault slip 0246810 time after earthquake (years) 6 (c) 0.06 4 0.04 surface displacement (m)

2 surface 0.02 displacement (m) 0 0 0 200 400 600 800 1000 01234 time after earthquake (years) time after earthquake (months)

Fig. 18. (a) Displacement at Earth's surface 20 km from the fault. Postseismic displacement at the same location for (b) 10 years and (c) 4 months after an earthquake. The solid lines show total displacement, and the dashed lines show the displacement resulting from fault slip alone. The diﬀerences in lower crustal deformation style as a function of geotherm lead to subtle diﬀerences in displacement time series that might be measurable.

247 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

(a) 20 K/km (d) 20 K/km 1 5 years 2 10 years 1/4 T L 3/4 T L 0.5 v/v 1 v/v

0 0 0 1020304050 0246810 (b) 25 K/km (e) 25 K/km 1

L L 0.5 v/v 1 v/v

0 0 0 1020304050 0246810 (c) 30 K/km (f) 30 K/km 1

2 L L 0.5 v/v 1 v/v

0 0 0 1020304050 0246810 distance from fault (km) distance from fault (km)

Fig. 19. Surface velocity predictions for each simulation at selected points in time throughout the interseismic period: 5 years after the earthquake (light green), 10 years after the earthquake (blue-green), 25% of the way through the interseismic period (dark blue), and 75% of the way through (black). The solid lines show total displacement and the dashed lines show the displacement resulting from fault slip alone, with their difference being from bulk viscous flow. T indicates the recurrence interval for each simulation. In contrast with bulk viscous flow, fault slip is more significant closer to the fault and over shorter timescales. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

6. Conclusions term in the overall force balance of the lithosphere, and as a result the integrated force on the fault is primarily balanced by shear stresses on In conclusion, we have developed a numerical method for simu- vertical planes parallel to the fault. As a result, and because the ductile lating earthquake cycles with rate-and-state friction on a strike-slip fault root experiences higher strain rates than the region far from the fault and a power-law viscoelastic off-fault rheology. We investigate the fault at the same depth, unrealistically high stresses develop in the interaction between fault slip and bulk viscous flow in a region similar upper crust far from the fault. to the Mojave Desert in Southern California, representing the crust with Resolving the crustal stress problem is a high priority, and we close experimentally-derived parameters for quartz-diorite and the upper by mentioning several ideas that could be tested in future studies using mantle with parameters for olivine. The three geotherms considered various extensions of the modeling framework that we have introduced produce very different deformation styles in the lower crust and upper here. By extending the current simulation capabilities to account for mantle, ranging from significant fault creep at depth to purely bulk off-fault plasticity, as in Erickson et al. (2017), would allow us to in- viscous flow. Despite the differences in lower crustal deformation style, vestigate if distributed plastic deformation or the development of ad- each simulation has almost identical recurrence interval, nucleation ditional subparallel strike-slip faults might relieve these stresses. The depth, down-dip coseismic slip limit, and total coseismic surface slip. high upper-crustal stresses might also be reduced by additional weak- This is probably because aseismic slip directly below the seismogenic ening mechanisms in either the seismogenic zone (e.g., dynamic zone occurs in a similar manner for all cases considered, in contrast to weakening) or in the ductile fault root (e.g., grain size reduction or the pronounced variability of the deeper deformation mechanism. thermo-mechanical feedback). Finally, it is also possible that for a Despite these similarities, the predicted postseismic deformation varies stronger (more mafic) lower crust, the basal tractions would play a between the simulations in a manner that might be used to distinguish more significant role in the force balance than suggested by our simu- the deformation process at depth. lations. Future studies exploring these possibilities and processes are Additionally, the simulations make different predictions for the certainly warranted. basal tractions on the lithosphere, with the 25 and 30 K/km simulations predicting that the crust drags the mantle at all distances from the fault, and the 20 K/km simulation predicting this at > 10 km from the fault, Acknowledgements and the reverse within 10 km of the fault. Basal tractions are a minor This study was supported by the U.S. Geological Survey under Grant

248 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

No. G17AP00013 and theSouthern California Earthquake Center discussions about lithospheric structure; Gary Mavko for discussions (Contribution No. 7272). SCEC is funded by NSF Cooperative about rheology and mixing laws; and Paul Segall for discussions about Agreement EAR-1033462 and USGS Cooperative Agreement crustal deformation. We acknowledge invaluable suggestions from two G12AC20038. We thank Brittany Erickson, Kenneth Duru, and Ken anonymous reviewers, which helped improve our discussion of crustal Mattsson for assistance with the numerics; Simon Klemperer for stresses.

Appendix A. Spatial discretization

The governing equations are discretized using summation-by-parts (SBP) finite difference operators for variable coefficient problems (Mattsson and Nordström, 2004; Mattsson, 2012). Boundary conditions are enforced weakly using simultaneous approximation terms (SAT), which penalize boundary grid data rather than overwriting grid data with the boundary conditions (injection, or strong enforcement of boundary conditions) (Mattsson et al., 2009). To prove that the numerical algorithm is stable, it is sufficient to prove that the semi-discrete energy dissipates at least as quickly as the energy of the continuum problem for homogeneous boundary conditions. In this Appendix, we first transform the governing equations to allow for variable grid spacing. Then, we describe the spatial discretization for these transformed equations. Finally, we prove stability of the spatial discretization by solving for the continuum and semi-discrete energy balance equations.

A.1. Variable grid spacing

* To ensure the accuracy of the numerical solution, both critical grid spacings, h (Eq. (20)) and Lb (Eq. (21)), must be resolved. These critical length scales require the use of a small grid spacing in the vicinity of the velocity-weakening regions of the fault, but elsewhere in the domain the grid spacing is permitted to be much larger. To allow for variable grid spacing, we implement a coordinate transform from curvilinear coordinates (y,z) to Cartesian coordinates (q,r) where (q,r) ∈ [0,1]. The two coordinates are transformed independently, such that y=y(q) and z=z(r), each transformation is invertible, and each has the property dq/dy > 0 and dr/dz > 0. An example is shown in Fig. 20. The static equilibrium equation, the coordinate transformed form of Eq. (1),is

q ⎛∂z ⎞ ∂ ⎛ ∂ ∂u V ⎞ ⎛ ∂y ⎞ ∂ ⎛ ∂r ∂u V ⎞ ⎜⎟⎜⎟μ − μγxy + μ − μγxz = 0 ⎝ ∂rq⎠ ∂ ⎝ ∂y ∂q ⎠ ⎝ ∂qr⎠ ∂ ⎝ ∂z ∂r ⎠ (A.1) and the strain rates are

V V dγxy ∂q dγ −−11⎛ ∂u V ⎞ xz ⎛ ∂r ∂u V ⎞ = ημ⎜⎟− γxy , = ημ − γxz . dt ⎝ ∂y ∂q ⎠ dt ⎝∂z ∂r ⎠ (A.2) The coordinate transformation does not change the mechanical energy balance, but does change the semi-discrete energy balance.

In the simulations in this paper, we use the coordinate transform y(q)=Ly sinh(10q)/sinh(10). For the z(r), we use the grid spacing and transformation shown in Fig. 21.

Fig. 20. Curvilinear coordinate transform from (y,z) with variable grid spacing to (q,r) with constant grid spacing.

249 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256 (a) (b) 0 0

10 10

20 20 z (km) depth (km)

30 30

40 40 0 0.2 0.4 0 0.5 1 z (km) r

Fig. 21. (a) Grid spacing Δz as a function of depth. (b) Coordinate transformation z(r), the integral of the function plotted in (a).

A.2. Semi-discrete governing equations

In this section, we describe the use of SBP finite difference operators to spatially discretize the governing equations. Let the coordinate grid be defined by qi ==…=−iqΔ, i 1, , Nqq , Δ q 1/( N 1) (A.3) rjrjjrr==…=−Δ, 1, , N , Δ r 1/( N 1) (A.4) where Nq is the number of grid points in the q-direction, Nr is the number of grid points in the r-direction. The grid function uij is a vector of length T NqNr and is u =…()uu11,,, 12 uNNqr . Denote the boundary conditions as u (0,zt , )= bL ( zt , ) (A.5) u (,,)LztyR= bzt (,) (A.6)

σxz (,0,)yt= 0 (A.7)

σxz(,yL z , t )= 0. (A.8)

()qμ The spatial derivative operators are denoted D ≈ ∂ and Dμy ≈ ∂ ∂q ∂ . Details of the first derivative operator are in Mattsson and 1 ∂q 2 ∂q ()∂yq∂ Nordström (2004), and of the second derivative with variable coefficients in Mattsson (2012), and we summarize only the key points here. We fi (qμy ) speci cally use the narrow stencil, fully compatible form of D2 . The one-dimensional form of the second derivative operator is ()μq y −1()μqy DHMμqBD2 =−() +y 1 , (A.9)

()μqyyT ()μq M =+DHμqD1 y 1 R , (A.10)

−1 dq q =≈()Dyq y dy (A.11)

B =−diag( 1,0,0, … ,0,1) (A.12) where μ is a diagonal matrix whose nonzero entries correspond to the values of the shear modulus at every point in the domain, H is a quadrature operator used for defining inner products and norms and is diagonal in the form of the SBP operators used in this paper, qy is a diagonal matrix approximating the Jacobian of the coordinate transform, and R is an additional energy term which vanishes with grid refinement. The matrices μ and μq qy are symmetric positive definite, and R, H, and M y are symmetric semi-positive definite. The two-dimensional operators, denoted in bold, are formed from the one-dimensional operators by applying the Kronecker product with the identity matrices Iy and Iz, such that

250 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

dz −1 ⎛ dr ⎞ zrrz=⊗IDzqr() ≈⎛ ⎞ , =≈zr ⎝ dr ⎠ ⎝dz ⎠

⎛ dy ⎞ −1 ⎛ dq ⎞ yqqy=⊗≈()Dyqr I ⎜⎟ , =≈yq ⎜⎟ ⎝ dq ⎠ ⎝ dy ⎠ HHHqqrrqr=⊗HI,, =⊗ IH =⊗ HH qr

Dqqr=⊗DI,

Drq=⊗ID r

()μμrrzz−1 () DMrr =⊗(IHq rr )( − +μrz(IBIDqrqr ⊗ )( ⊗ )) ()μμrzzT r Mrr =⊗()()()IDIHq r qr ⊗μ z ID qr ⊗ +Rr ()μμqqyy−1 () DMqq =⊗−+(HIq z )( q μqy (BIDIqrqr ⊗ )( ⊗ ))

()μμqyyT q Mqq =⊗()()().DIHIq rq ⊗ rμ y DI q ⊗+ r Rq μq y μrz The form of Rq and Rr are given in Mattsson (2012). The matrices

ELz=…⊗diag(1, 0, ,0) I

ERz=…⊗diag(0, ,0, 1) I

ETy=⊗I diag(1, … ,0, 0)

EBy=⊗I diag(0, 0, … ,1) pick out the left, right, top, and bottom boundaries, respectively. We also use the notation

BEEBEEqLRrTB=− +, =− + .

The semi-discrete form of Hooke's Law (Eq. (2))is

V σxy =−μ ()qDy quγxy (A.13)

V σxz =−μ ()rDz r uγxz . (A.14) The semi-discrete form of the static equilibrium equation (Eq. (A.1))is

()μq y V ()μrz V 0 =−+++−++zDr ()qq uγ Dqμμxy pLR p yDq ()rr uγ Dr xz pBT p, (A.15) where

−1 T pL =+Hqq ((αubμμyy qBDEqq ))(LL −) (A.16)

−1 T pR =+Hqq ((αubμμyy qBDEqq ))(RR −) (A.17)

−1 pT =−HEr Tr[(μ rDz uγxz )] (A.18)

−1 pB =−HEr Br[(μ rDz uγ− xz )]. (A.19)

The penalty terms pL, pR, pT, and pB weakly enforce the left, right, top, and bottom boundary conditions, Eqs. (A.5)–(A.8), respectively. The semi- discrete strain rates, corresponding to Eq. (3), are dγV xy −1 V −1 −1 −1 −1 =−+−−−ημ()qDquγ Hημ qELL ()(ub Hημ qERRub) dt yyxy q q y (A.20) dγV xz −1 V =−ημ(),rDr uγ dt z xz (A.21) where η is a diagonal, positive deﬁnite matrix whose nonzero entries correspond to the viscosity at every point in the domain. The last two terms in

Eq. (A.20) are penalty terms introduced to ensure stability for Dirichlet boundary conditions at y=0 and y=Ly.

A.3. Proof of stability

The stability proof makes use the property of the Kronecker product (A ⊗ B)(C ⊗ D)=(AC) ⊗ (BD) and the property of the ﬁrst derivative

−1 T DH1 =−+( DHB1 ) (A.22) −1 where D1 can be applied in either the q or r direction, and H, H , B should be interpreted appropriately. We ﬁrst consider the energy balance of the continuum problem. To compute the mechanical energy balance equation, multiply the static equilibrium equation (Eq. (1))byu̇and integrate over the domain. The energy balance equation is

L L d Lz y Ly z E=+−∫∫ uσdżxy uσdẏxz Φ dt 0 y==0 0 z 0 (A.23)

LL L Ly L Lz 1 zz1 2 z1 y = −+τVdz η τV dz + σxy V L dz +uσ̇xz dy −Φ ∫∫00rad ∫0 ∫0 2 2 2 y=0 z=0 (A.24)

251 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256 where the elastic strain energy is

LL 1 yz22 E =+∫∫μσ()xy σ xz dydz 2 00 (A.25) and the rate of energy dissipation from viscous ﬂow is

LL yz−12 22 Φ()=+η μ σxy σ xz dydz. ∫∫00 (A.26) In Eq. (A.24), the ﬁrst term on the right-hand side is the frictional heat dissipation, the second is the energy that ﬂows into the fault via seismic radiation (as idealized with the radiation damping approximation), and the third and fourth are the rate of work on the system by the boundary tractions. For homogeneous boundary conditions, the boundaries do not contribute energy to the system. To compute the semi-discrete energy balance equation, left multiply the static equilibrium equation (Eq. (A.15))byuṪH. The elastic energy is

T 1 1 Dyuγ− V Dyuγ− V 1 E =−()()DzHyrDzuγV T μ uγ−+V ⎡ q q xy⎤ A⎡ q q xy⎤ + uTHyR()μrz u r r xz q zrr xz ⎢ ⎥ ⎢ ⎥ q q r 2 2 ⎣ u ⎦ ⎣ u, ⎦ 2 (A.27) where

μμrHzzr − qy Bq  ⎡ ⎤ A = ⎢ ()μq ⎥. −−+μμqB HzRy α H q() E E ⎣⎢ y qrr q rLRy ⎦⎥ (A.28) In Eq. (A.27), the ﬁrst term and the term resulting from the upper left block A correspond to the mechanical elastic energy (Eq. (A.25)). Of the ()μq (μrz) y ﬁ remaining terms, those with Rr and Rq are arti cial numerical energy terms that vanish as the grid spacing converges. The rest, resulting from the remaining elements in A, are the result of the weakly enforced boundary conditions. The energy balance equation for homogeneous boundary conditions is

d V T −12 V E =−()Dzr uγ −ημ rHyDz ()r uγ − + dt rzxz q r xz T −12 V ⎡ημqHzy r r 0 ⎤ V ⎡Dyquγ− q xy⎤ ⎡Dyquγ− q xy⎤ ⎢ −12 ⎥B ⎢ u ⎥ 0 ημqHzr r ⎢ u ⎥ ⎣ ⎦ ⎣ y ⎦ ⎣ ⎦ (A.29) where

⎡− HBqq⎤ B = . ⎢ BHEE−+−1 ()⎥ ⎣ q q LR⎦ (A.30) In Eq. (A.29), the first term and the term resulting from the upper left block B correspond to the continuum energy balance (Eq. (A.23)) and the remaining terms are the result of the weakly enforced boundary conditions. The requirement that the semi-discrete energy dissipate at least as quickly as the continuous energy is met if A is symmetric positive semi-definite and B is symmetric negative semi-definite. B meets this requirement by construction, and A does if the SAT penalty weight α is less than or equal to fi −1 −1 α ≤ Δ α ≤ Δ Δ the rst diagonal entry of Hq , that is α ≤ ()Hq 00. For second order accuracy, 2/ q, and for fourth order (48/17)/ q, where q is the grid spacing in the q-direction (Mattsson, 2012; Mattsson et al., 2009).

Appendix B. Veriﬁcation of implementation

To verify the accuracy of the numerical algorithm for the static equilibrium equation (Eq. (1)) and viscous strain rate (Eq. (3)) equations, we use the method of manufactured solutions (MMS). For the linear Maxwell simulations, we use a domain of size [0,2π km]×[0,1 km] and integrate in time over 200 years. We impose the material properties μ =+sin (yz )sin ( ) 30GPa, (B.1)

η =+cos (yz )cos ( ) 2× 1019 Pa s. (B.2) We also impose the displacement and viscous strain ﬁelds

ûcos()sin()(=−yzee−−−tT///123 tT+ e tT)m, (B.3)

−1 TT1 (sinsin)− y z −1 V̂ max −−tT/ 1 Tmax t γxy= −1 ()ee− TT1 max − 1 −1 TT(sinsin)− y z −1 2 max −−tT/ 2 Tmax t − −1 ()ee− TT2 max − 1 −1 TT(sinsin)− y z −1 3 max −−tT/ 3 Tmax t + −1 (),ee− TT3 max − 1 (B.4)

252 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

−1 TT1 (cos y cos z ) −1 V̂ max −−tT/ 1 Tmax t γxz= −1 ()ee− TT1 max − 1 −1 TT(cos y cos z ) −1 2 max −−tT/ 2 Tmax t − −1 ()ee− TT2 max − 1 −1 TT(cos y cos z ) −1 3 max −−tT/ 3 Tmax t + −1 (),ee− TT3 max − 1 (B.5) where Tmax =η/μ, and T1 =60s, T2 =1 year, and T3 =100 years are timescales which are representative of the coseismic, postseismic and interseismic periods. Boundary conditions are set by u (0,zt , )= û(0, zt , ) (B.6) u (,,)Lztyy= û(,,) Lzt (B.7)

⎛ ∂û(0,zt , ) V̂⎞ σxz (,0,)ytμ= ⎜⎟− γ ∂y xy ⎝ ⎠ z=0 (B.8)

⎛ ∂û(0,zt , ) V̂⎞ σxz(,yL z , t )= μ⎜⎟− γ . ∂y xy ⎝ ⎠ z=1 (B.9) ̂ ̂ The static equilibrium (Eq. (1)) and viscous strain rate (Eq. (3)) equations are not solved exactly by û, γxy, and γxy, so source terms are added to them. For the power-law viscoelastic simulations, we use the same spatial domain and time interval and impose the material properties μ =+sin (yz )sin ( ) 30 GPa, (B.10)

Ayz=+cos ( )cos ( ) 398 MPa s, (B.11)

Q =+sin (yz )sin ( ) 135 kJ mol-1, (B.12)

Tyz=+sin ( )cos ( ) 800 K, (B.13) nyz=+cos ( )sin ( ) 3. (B.14) ̂ ̂ Again, the static equilibrium (Eq. (1)) and viscous strain rate (Eq. (3)) equations are not solved exactly by û, γxy, and γxy, so source terms are added to them. We measure convergence of the numerical solution u* to the analytical solution û with the relative error

||u* − û||2 Error2 = . ||û||2 (B.15) For the second derivative for fully compatible SBP operators, if the order of accuracy is denoted p in the interior and p/2 on the boundary, then the global convergence rate is expected to be p/2+1 (Mattsson, 2012). This means that 2nd order accurate (in the interior) operators will converge at a rate of at least 2, and 4th order at a rate of at least 3. The results of this MMS test are shown in Fig. 22. Note that the 4th order accurate simulation converges at a slope of 4. (a) (b) 10-6 10-6

10-8 10-8

) -10 -10

2 10 10

error (L 10-12 10-12

10-14 10-14

10-16 10-16 10-4 10-2 100 10-4 10-2 100 grid spacing grid spacing

Fig. 22. Results of the MMS convergence test on displacement for the linear (a) and power-law (b) viscoelastic rheologies. Blue lines are for second order accuracy and red for fourth. The black dashed lines indicate the slope of convergence, which is 2 for 2nd order and 4 for 4th order. (For interpretation of the references to color in this ﬁgure legend, the reader is referred to the web version of this article.)

253 K.L. Allison, E.M. Dunham Tectonophysics 733 (2018) 232–256

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